Question

As shown below, convex pentagon $ABCDE$ has sides $AB=3,$ $BC=4,$ $CD=6,$ $DE=3,$ and $EA=7.$ The pentagon is originally positioned in the plane with vertex $A$ at the origin and vertex $B$ on the positive $x$ -axis. The pentagon is then rolled clockwise to the right along the $x$ -axis. Which side will touch the point $x=2009$ on the $x$ -axis?

$\overline{AB}$

$\overline{BC}$

$\overline{CD}$

$\overline{DE}$

$\overline{EA}$

Solution:

The pentagon has perimeter $3+4+6+3+7=23.$ One full roll advances the contact point by $23,$ and $2009=23\cdot87+8.$

After $87$ rolls, vertex $A$ sits at $x=23\cdot87=2001$ and $B$ at $2004.$ Rolling further, $C$ touches at $2004+4=2008$ and $D$ at $2008+6=2014.$

Since $2009$ lies between $2008$ and $2014,$ side $\overline{CD}$ touches that point.

Thus, the correct answer is C.

2009 AMC 10B Problem 13

Solution:

Problem 13 in Other Years